In planetary motion the areal velocity of position vector of a planet depends on angular velocity (m) and the distance of the planet from sup (r). If so the correct relation for areal velocity is

Options

A. `(dA)/(dt) prop omega r`
B. `(dA)/(dt) prop omega^2 r`
C. `(dA)/(dt) prop omega r^2`
D. `(dA)/(dt) prop sqrt(omega r)`

Answer

To solve the problem of finding the relation for areal velocity in planetary motion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Areal Velocity**:
Areal velocity is defined as the area swept out by the position vector of a planet per unit time. Mathematically, it can be expressed as:
\[ \text{Areal velocity} = \frac{dA}{dt} \]
where \(dA\) is the differential area swept out in a small time interval \(dt\).

2. **Dimensional Analysis**:
The dimensions of area \(A\) are \(L^2\) (length squared), and the dimensions of time \(t\) are \(T\). Therefore, the dimensions of areal velocity are:
\[ \frac{dA}{dt} \sim \frac{L^2}{T} \quad \text{(dimensions of areal velocity)} \]

3. **Identifying Variables**:
According to the problem, the areal velocity depends on:
- Angular velocity (\(\omega\)), which has dimensions of \(T^{-1}\) (since it is measured in radians per second).
- Distance from the Sun (\(r\)), which has dimensions of \(L\).

4. **Assuming a Relation**:
We can assume a relation of the form:
\[ \frac{dA}{dt} \propto \omega^a \cdot r^b \]
where \(a\) and \(b\) are exponents that we need to determine.

5. **Setting Up the Dimensional Equation**:
From the proportionality, we can write the dimensions:
\[ [\frac{dA}{dt}] = [\omega^a] \cdot [r^b] \]
Substituting the dimensions we have:
\[ \frac{L^2}{T} = (T^{-1})^a \cdot (L)^b \]
This simplifies to:
\[ \frac{L^2}{T} = L^b \cdot T^{-a} \]

6. **Equating Dimensions**:
Now we can equate the dimensions on both sides:
- For length \(L\):
\[ 2 = b \quad \Rightarrow \quad b = 2 \]
- For time \(T\):
\[ -1 = -a \quad \Rightarrow \quad a = 1 \]

7. **Final Relation**:
Substituting back the values of \(a\) and \(b\) into our assumed relation, we get:
\[ \frac{dA}{dt} \propto \omega^1 \cdot r^2 \]
Therefore, the relation for areal velocity is:
\[ \frac{dA}{dt} \propto \omega r^2 \]

8. **Conclusion**:
The correct relation for areal velocity in planetary motion is:
\[ \frac{dA}{dt} = k \cdot \omega r^2 \]
where \(k\) is a proportionality constant.